Re-Phase Vol.3.2: Coding-Theoretic Re-Phase — Encoding Relations, Syndrome Fibers, Cosets, and Structural Recoverability

Description This volume develops the coding-theoretic specialization of the Re-Phase framework. The central viewpoint is that a code does not merely map information into another space. Rather, it defines compatibility fibers. Encoding is treated first as a relation between a source-side information space and an encoded-word space, while decoding is interpreted as representative selection from a fiber cut out by a received or syndrome condition. At the relation level, an encoding relation (\Gamma \subset E_p \times E_q) induces decoding fibers[A_q={p\in E_p\mid (p,q)\in\Gamma}.]A hard channel relation (N\subset W\times Y) composes with an encoding relation (\Gamma_{\mathrm{enc}}\subset E_p\times W) to form a source-to-received compatibility relation. At the linear syndrome level, a syndrome map[H:X\to\Sigma]cuts out admissible syndrome fibers[A_\sigma=H^{-1}(\sigma).]When nonempty, these fibers are affine cosets[A_\sigma=x_0+\ker H.]Writing (C=\ker H), codes appear as syndrome indistinguishability kernels and syndrome fibers appear as admissible cosets. The main distinction of this volume is that decoding and recoverability are separated. A hard decoder selects a representative (D(\sigma)\in A_\sigma), whereas Re-Phase recoverability evaluates the feature image of the whole fiber:[h(A_\sigma)\subset B.]Thus, representative selection is not the same as recoverability over the fiber. The volume also shows that, for linear feature maps, unique feature recoverability over syndrome fibers is characterized by[C\subseteq\ker h.]Equivalently, recoverable linear features factor through the syndrome map. In this sense, recoverable linear features are syndrome functions. This work does not attempt to reproduce coding theory or replace standard syndrome decoding, coset decoding, list decoding, channel coding, or probabilistic decoding. Instead, it repositions coding-theoretic structures inside the Re-Phase grammar of compatibility relations, admissible fibers, representative selection, and structural recoverability. Soft-decision decoding, probabilistic channels, ML/MAP decoding, capacity theory, decoding algorithms, and delayed-closure symbolic sequences are deferred to later layers.